, 55:19 | Cite as

A posteriori error estimates for fully discrete schemes for the time dependent Stokes problem

  • E. BänschEmail author
  • F. Karakatsani
  • C. G. Makridakis


This work is devoted to a posteriori error analysis of fully discrete finite element approximations to the time dependent Stokes system. The space discretization is based on popular stable spaces, including Crouzeix–Raviart and Taylor–Hood finite element methods. Implicit Euler is applied for the time discretization. The finite element spaces are allowed to change with time steps and the projection steps include alternatives that is hoped to cope with possible numerical artifices and the loss of the discrete incompressibility of the schemes. The final estimates are of optimal order in \(L^\infty (L^2) \) for the velocity error.


A posteriori error estimators Time dependent Stokes Reconstruction Adaptivity Mesh change Crouzeix–Raviart element 

Mathematics Subject Classification

65M15 65M50 65N15 


  1. 1.
    Akrivis, G., Makridakis, C., Nochetto, R.H.: A posteriori error estimates for the Crank–Nicolson method for parabolic equations. Math. Comp. 75, 511–531 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Akrivis, G., Makridakis, C., Nochetto, R.H.: Optimal order a posteriori error estimates for a class of Runge–Kutta and Galerkin methods. Numer. Math. 114, 133–160 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Alnaes, M.S., Blechta, J., Hake, J., Johansson, A., Kehlet, B., Logg, A., Richardson, C., Ring, J., Rognes, M., Wells, G.N.: The FEniCS project version 1.5. Archive of Numerical Software. 3(100) (2015)Google Scholar
  4. 4.
    Bänsch, E.: Local mesh refinement in 2 and 3 dimensions. Impact Comput. Sci. Eng. 3(3), 181–191 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bänsch, E.: Simulation of instationary, incompressible flows. In: Proceedings of the Algoritmy’97 Conference on Scientific Computing (Zuberec), vol. 67, pp. 101–114 (1998)Google Scholar
  6. 6.
    Bänsch, E., Karakatsani, F., Makridakis, C.: A posteriori error control for fully discrete Crank–Nicolson schemes. SIAM J. Numer. Anal. 50(6), 2845–2872 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bänsch, E., Karakatsani, F., Makridakis, C.: The effect of mesh modification in time on the error control of fully discrete approximations for parabolic equations. Appl. Numer. Math. 67, 35–63 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bänsch, E., Morin, P., Nochetto, R.H.: An adaptive Uzawa FEM for the Stokes problem: convergence without the inf-sup condition. SIAM J. Numer. Anal. 40(4), 1207–1229 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bänsch, E., Schmidt, A.: Simulation of dendritic crystal growth with thermal convection. Interfaces Free Bound. 2(1), 95–115 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bernardi, C., Sayah, T.: A posteriori error analysis of the time-dependent Stokes equations with mixed boundary conditions. IMA J. Numer. Anal. 35(1), 179–198 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bernardi, C., Verfürth, R.: A posteriori error analysis of the fully discretized time-dependent Stokes equations. M2AN. Math. Model. Numer. Anal. 38(3), 437–455 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Besier, M., Wollner, W.: On the pressure approximation in nonstationary incompressible flow simulations on dynamically varying spatial meshes. Int. J. Numer. Methods Fluids 69, 1045–1064 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Brenner, A., Bänsch, E., Bause, M.: A priori error analysis for finite element approximations of the Stokes problem on dynamic meshes. IMA J. Numer. Anal. 34(1), 123–146 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Springer, New York (1994)CrossRefzbMATHGoogle Scholar
  15. 15.
    Chatzipantelidis, P., Makridakis, C., Plexousakis, M.: A-posteriori error estimates for a finite volume method for the Stokes problem in two dimensions. Appl. Numer. Math. 46(1), 45–58 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Crouzeix, M., Raviart, P.-A.: Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I. Rev. Française. Automat. Informat. Recherche Opérationnelle Sér. Rouge 7(R–3), 33–75 (1973)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Dari, E., Durán, R., Padra, C.: Error estimators for nonconforming finite element approximations of the Stokes problem. Math. Comp. 64(211), 1017–1033 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Dautray, R., Lions, J.-L.: Mathematical Analysis and Numerical Methods for Science and Technology, vol. 6. Springer, Berlin (1993). Evolution problems. II, With the collaboration of Claude Bardos, Michel Cessenat, Alain Kavenoky, Patrick Lascaux, Bertrand Mercier, Olivier Pironneau, Bruno Scheurer and Rémi SentiszbMATHGoogle Scholar
  19. 19.
    Dupont, T.: Mesh modification for evolution equations. Math. Comput. 39, 85–107 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Girault, V., Raviart, P.-A.: Finite Element Methods for Navier–Stokes Equations. Theory and Algorithms, Volume 5 of Springer Series in Computational Mathematics. Springer, Berlin (1986)zbMATHGoogle Scholar
  21. 21.
    Hannukainen, A., Stenberg, R., Vohralík, M.: A unified framework for a posteriori error estimation for the Stokes problem. Numer. Math. 122(4), 725–769 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Heywood, J.G., Rannacher, R.: Finite element approximation of the nonstationary Navier–Stokes problem. I. Regularity of solutions and second-order error estimates for spatial discretization. SIAM J. Numer. Anal. 19(2), 275–311 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Heywood, J.G., Rannacher, R.: An analysis of stability concepts for the Navier–Stokes equations. J. Reine Angew. Math. 372, 1–33 (1986)MathSciNetzbMATHGoogle Scholar
  24. 24.
    John, V.: A posteriori \(L^2\)-error estimates for the nonconforming \(P_1/P_0\)-finite element discretization of the Stokes equations. J. Comput. Appl. Math. 96(2), 99–116 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Karakashian, O., Makridakis, C.: A space–time finite element method for the nonlinear Schrödinger equation: the discontinuous Galerkin method. Math. Comp. 67(222), 479–499 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Karakatsani, F.: A posteriori error estimates for the fractional-step \(\vartheta \)-scheme for linear parabolic equations. IMA J. Numer. Anal. 32(1), 141–162 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Karakatsani, F., Makridakis, C.: A posteriori estimates for approximations of time-dependent Stokes equations. IMA J. Numer. Anal. 27(4), 741–764 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Lakkis, O., Makridakis, C.: Elliptic reconstruction and a posteriori error estimates for fully discrete linear parabolic problems. Math. Comp. 75(256), 1627–1658 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Makridakis, C., Nochetto, R.H.: Elliptic reconstruction and a posteriori error estimates for parabolic problems. SIAM J. Numer. Anal. 41(4), 1585–1594 (2003). (electronic)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Schmidt, A., Siebert, K.G.: Design of Adaptive Finite Element Software, Volume 42 of Lecture Notes in Computational Science and Engineering. Springer, Berlin (2005). The finite element toolbox ALBERTA, With 1 CD-ROM (Unix/Linux)Google Scholar
  31. 31.
    Taylor, C., Hood, P.: A numerical solution of the Navier–Stokes equations using the finite element technique. Int. J. Comput. Fluids 1(1), 73–100 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Temam, R.: Navier–Stokes Equations. Theory and Numerical Analysis. AMS Chelsea Publishing, Providence (2001). Reprint of the 1984 editionzbMATHGoogle Scholar
  33. 33.
    Verfürth, R.: A posteriori error estimators for the Stokes equations. Numer. Math. 55(3), 309–325 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Verfürth, R.: A posteriori error analysis of space–time finite element discretizations of the time-dependent Stokes equations. Calcolo 47(3), 149–167 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Verfürth, R.: A Posteriori Error Estimation Techniques for Finite Element Methods. Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford (2013)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Italia S.r.l., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Applied Mathematics IIIErlangenGermany
  2. 2.Department of Mathematics, Faculty of Science and Engineering, Thornton Science ParkUniversity of ChesterInce, ChesterUK
  3. 3.Modelling and Sc. Computing, DMAMUniversity of CreteHeraklionGreece
  4. 4.Institute for Applied and Computational Mathematics, FORTHHeraklionGreece
  5. 5.MPSUniversity of SussexBrightonUK

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